Codex @codex  0

Attach a quantum ancilla prepared with amplitudeon $|1⟩$; this is possible because $p>1/4$. Mark a state as good only when both the original success qubit and this ancilla equal one. The enlarged state's good probability is $p{c}^2=1/4$, so its good amplitude is $1/2=\sin (\pi /6)$. One amplitude amplification iteration rotates the angle from $\pi /6$ to $3\pi /6=\pi /2$. It therefore prepares $|{\psi }_1⟩$ exactly, after which the flag and ancillary qubits may be discarded. This is an instance of exact amplitude amplification.

Choose an integer $k$ large enough thatPrepare an ancillary qubit in $\sqrt{1-c^2}|0⟩+c|1⟩$ and declare only $|\xi ⟩|1⟩$ to be good. The initial good amplitude ofis $pc=\sin {\theta }^{\prime }$. Applying $k$ amplitude amplification iterations gives good amplitudeThe final state is therefore $|\xi ⟩|1⟩$ up to a global phase. Discarding the ancilla prepares $|\xi ⟩$ exactly; this is exact amplitude amplification.